Page 553 - Handbook Of Integral Equations

11.4. Method of Fredholm Determinants

11.4-1. A Formula for the Resolvent
A solution of the Fredholm equation of the second kind

b
y(x) – λ K(x, t)y(t) dt = f(x), a ≤ x ≤ b, (1)
a
is given by the formula
b

y(x)= f(x)+ λ R(x, t; λ)f(t) dt, a ≤ x ≤ b, (2)
a
where the resolvent R(x, t; λ)isdefined by the relation

D(x, t; λ)
R(x, t; λ)= , D(λ) ≠ 0. (3)
D(λ)
Here D(x, t; λ) and D(λ) are power series in λ,

∞ n ∞ n
(–1) n (–1) n
D(x, t; λ)= A n (x, t)λ , D(λ)= B n λ , (4)
n! n!
n=0 n=0
with coefficients defined by the formulas
K(x, t) K(x, t 1 ) ···

b b K(t 1 , t) K(t 1 , t 1 ) ··· K(x, t n)
A 0 (x, t)= K(x, t), A n (x, t)= ··· . . . K(t 1 , t n) dt 1 ...dt n , (5)
.
. . . .
a a . . . .
K(t n, t) K(t n, t 1 ) ··· K(t n, t n)

n
K(t 1 , t 1 ) K(t 1 , t 2 ) ··· K(t 1 , t n)

b b K(t 2 , t 1 ) K(t 2 , t 2 ) ···

B 0 =1, B n = ··· . . . K(t 2 , t n) dt 1 ...dt n ; n = 0,1,2, ... (6)
.
. . . .
a a . . . .

K(t n, t 1 ) K(t n, t 2 ) ··· K(t n, t n)
n
The function D(x, t; λ) is called the Fredholm minor and D(λ) the Fredholm determinant. The
series (4) converge for all values of λ and hence define entire analytic functions of λ. The resolvent
R(x, t; λ) is an analytic function of λ everywhere except for the values of λ that are roots of D(λ).
These roots coincide with the characteristic values of the equation and are poles of the resolvent
R(x, t; λ).
Example 1. Consider the integral equation
1

t
y(x) – λ xe y(t) dt = f(x), 0 ≤ x ≤ 1, λ ≠ 1.
0
We have
t t t
xe
xe 1
1 xe xe 1 1
t t 1 xe 2
t
t
t
A 0 (x, t)= xe , A 1 (x, t)= dt 1 =0, A 2 (x, t)= t 1 e t t 1 e 1 t 1 e 2 dt 1 dt 2 =0,
t 1 e t t
0 t 1 e 1 0 0 t t t
t 2 e t 2 e 1 t 2 e 2
since the determinants in the integrand are zero. It is clear that the relation A n(x, t) = 0 holds for the subsequent coefficients.
Let us find the coefficients B n:
t
1 1 1 1 t 1 e 1 t

t 1 e 2
t 1
B 1 = K(t 1 , t 1 ) dt 1 = t 1 e dt 1 =1, B 2 = dt 1 dt 2 =0.
t
t 2 e 1 t
0 0 0 0 t 2 e 2
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 536